Problem: If $(x + y)^2 = 1$ and $xy = -4$, what is the value of $x^2 + y^2$?
Solution: We see that $(x + y)^2 = (x^2 + y^2) + 2xy = 1$. We want to find $x^2 + y^2$ and are given $xy = -4$. So, $x^2 + y^2 + 2xy = x^2 + y^2 + 2(-4) = 1$. It follows that $x^2 + y^2 = \boxed 9$.